List Edge-Colorings of Series-Parallel Graphs

It is proved that for every integer $k\ge3$, for every (simple) series-parallel graph $G$ with maximum degree at most $k$, and for every collection $(L(e):e\in E(G))$ of sets, each of size at least $k$, there exists a proper edge-coloring of $G$ such that for every edge $e\in E(G)$, the color of $e$ belongs to $L(e)$.

Download Full-text

On the Adjacent Vertex Distinguishing Proper Edge Colorings of Several Classes of Complete 5-Partite Graphs

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.333-335.1452 ◽

2013 ◽

Vol 333-335 ◽

pp. 1452-1455

Author(s):

Chun Yan Ma ◽

Xiang En Chen ◽

Fang Yang ◽

Bing Yao

Keyword(s):

Chromatic Number ◽

Edge Coloring ◽

Adjacent Vertex ◽

Chromatic Numbers ◽

Edge Colorings ◽

Proper Edge Coloring ◽

Minimum Number

A proper $k$-edge coloring of a graph $G$ is an assignment of $k$ colors, $1,2,\cdots,k$, to edges of $G$. For a proper edge coloring $f$ of $G$ and any vertex $x$ of $G$, we use $S(x)$ denote the set of thecolors assigned to the edges incident to $x$. If for any two adjacent vertices $u$ and $v$ of $G$, we have $S(u)\neq S(v)$,then $f$ is called the adjacent vertex distinguishing proper edge coloring of $G$ (or AVDPEC of $G$ in brief). The minimum number of colors required in an AVDPEC of $G$ is called the adjacent vertex distinguishing proper edge chromatic number of $G$, denoted by $\chi^{'}_{\mathrm{a}}(G)$. In this paper, adjacent vertex distinguishing proper edge chromatic numbers of several classes of complete 5-partite graphs are obtained.

Download Full-text

The Dominator Edge Coloring of Graphs

Mathematical Problems in Engineering ◽

10.1155/2021/8178992 ◽

2021 ◽

Vol 2021 ◽

pp. 1-7

Author(s):

Minhui Li ◽

Shumin Zhang ◽

Caiyun Wang ◽

Chengfu Ye

Keyword(s):

Chromatic Number ◽

Edge Coloring ◽

Simple Graph ◽

Edge Colorings ◽

Color Class ◽

Proper Edge Coloring ◽

Minimum Number ◽

Relationship Of ◽

The Relationship

Let G be a simple graph. A dominator edge coloring (DE-coloring) of G is a proper edge coloring in which each edge of G is adjacent to every edge of some color class (possibly its own class). The dominator edge chromatic number (DEC-number) of G is the minimum number of color classes among all dominator edge colorings of G , denoted by χ d ′ G . In this paper, we establish the bounds of the DEC-number of a graph, present the DEC-number of special graphs, and study the relationship of the DEC-number between G and the operations of G .

Download Full-text

KŐNIG’S LINE COLORING AND VIZING’S THEOREMS FOR GRAPHINGS

Forum of Mathematics Sigma ◽

10.1017/fms.2016.22 ◽

2016 ◽

Vol 4 ◽

Cited By ~ 1

Author(s):

ENDRE CSÓKA ◽

GÁBOR LIPPNER ◽

OLEG PIKHURKO

Keyword(s):

Maximum Degree ◽

Edge Coloring ◽

Classical Theorem ◽

Orbit Equivalence ◽

Bounded Degree ◽

Finite Graphs ◽

Edge Colorings ◽

Graph Limits ◽

Degree Graph ◽

Odd Cycles

The classical theorem of Vizing states that every graph of maximum degree $d$ admits an edge coloring with at most $d+1$ colors. Furthermore, as it was earlier shown by Kőnig, $d$ colors suffice if the graph is bipartite. We investigate the existence of measurable edge colorings for graphings (or measure-preserving graphs). A graphing is an analytic generalization of a bounded-degree graph that appears in various areas, such as sparse graph limits, orbit equivalence and measurable group theory. We show that every graphing of maximum degree $d$ admits a measurable edge coloring with $d+O(\sqrt{d})$ colors; furthermore, if the graphing has no odd cycles, then $d+1$ colors suffice. In fact, if a certain conjecture about finite graphs that strengthens Vizing’s theorem is true, then our method will show that $d+1$ colors are always enough.

Download Full-text

Restricted Extension of Sparse Partial Edge Colorings of Complete graphs

The Electronic Journal of Combinatorics ◽

10.37236/9552 ◽

2021 ◽

Vol 28 (2) ◽

Author(s):

Carl Johan Casselgren ◽

Lan Anh Pham

Keyword(s):

Complete Graph ◽

Edge Coloring ◽

Positive Answer ◽

Complete Graphs ◽

Edge Colorings ◽

Proper Edge Coloring

Given a partial edge coloring of a complete graph $K_n$ and lists of allowed colors for the non-colored edges of $K_n$, can we extend the partial edge coloring to a proper edge coloring of $K_n$ using only colors from the lists? We prove that this question has a positive answer in the case when both the partial edge coloring and the color lists satisfy certain sparsity conditions.

Download Full-text

ON PALETTE INDEX OF UNICYCLE AND BICYCLE GRAPHS

Proceedings of the YSU A: Physical and Mathematical Sciences ◽

10.46991/pysu:a/2019.53.1.003 ◽

2019 ◽

Vol 53 (1 (248)) ◽

pp. 3-12

Author(s):

A.B. Ghazaryan

Keyword(s):

Upper Bound ◽

Maximum Degree ◽

Edge Coloring ◽

Cyclomatic Number ◽

Proper Edge Coloring ◽

Minimum Number ◽

Delta G

Given a proper edge coloring $ \phi $ of a graph $ G $, we define the palette $ S_G (\nu, \phi) $ of a vertex $ \nu \mathclose{\in} V(G) $ as the set of all colors appearing on edges incident with $ \nu $. The palette index $ \check{s} (G) $ of $ G $ is the minimum number of distinct palettes occurring in a proper edge coloring of $ G $. In this paper we give an upper bound on the palette index of a graph G in terms of cyclomatic number $ cyc(G) $ of $ G $ and maximum degree $ \Delta (G) $ of $ G $. We also give a sharp upper bound for the palette index of unicycle and bicycle graphs.

Download Full-text

NEIGHBOR SUM DISTINGUISHING COLORING OF SOME GRAPHS

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830912500474 ◽

2012 ◽

Vol 04 (04) ◽

pp. 1250047 ◽

Cited By ~ 16

Author(s):

AIJUN DONG ◽

GUANGHUI WANG

Keyword(s):

Planar Graph ◽

Edge Coloring ◽

Proper Edge Coloring

A proper [k]-edge coloring of a graph G is a proper edge coloring of G using colors of the set [k] = {1, 2,…,k}. A neighbor sum distinguishing [k]-edge coloring of G is a proper [k]-edge coloring of G such that for each edge uv ∈ E(G), the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v. By ndiΣ(G), we denote the smallest value k in such a coloring of G. In this paper, we obtain that (1) ndiΣ(G) ≤ max {2Δ(G) + 1, 25} if G is a planar graph, (2) ndiΣ(G) ≤ max {2Δ(G), 19} if G is a graph such that mad(G) ≤ 5.

Download Full-text

A note on compact and compact circular edge-colorings of graphs

Discrete Mathematics & Theoretical Computer Science ◽

10.46298/dmtcs.431 ◽

2008 ◽

Vol Vol. 10 no. 3 (Graph and Algorithms) ◽

Author(s):

Dariusz Dereniowski ◽

Adam Nadolski

Keyword(s):

Approximation Algorithm ◽

Decision Problem ◽

Edge Coloring ◽

Exact Algorithm ◽

Weighted Graphs ◽

Edge Colorings ◽

Odd Cycle ◽

International Audience ◽

Np Complete ◽

Odd Cycles

Graphs and Algorithms International audience We study two variants of edge-coloring of edge-weighted graphs, namely compact edge-coloring and circular compact edge-coloring. First, we discuss relations between these two coloring models. We prove that every outerplanar bipartite graph admits a compact edge-coloring and that the decision problem of the existence of compact circular edge-coloring is NP-complete in general. Then we provide a polynomial time 1:5-approximation algorithm and pseudo-polynomial exact algorithm for compact circular coloring of odd cycles and prove that it is NP-hard to optimally color these graphs. Finally, we prove that if a path P2 is joined by an edge to an odd cycle then the problem of the existence of a compact circular coloring becomes NP-complete.

Download Full-text

Adjacent vertex distinguishing edge coloring of planar graphs without 3-cycles

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830920500354 ◽

2020 ◽

Vol 12 (04) ◽

pp. 2050035

Author(s):

Danjun Huang ◽

Xiaoxiu Zhang ◽

Weifan Wang ◽

Stephen Finbow

Keyword(s):

Planar Graph ◽

Planar Graphs ◽

Edge Coloring ◽

Discrete Math ◽

Adjacent Vertex ◽

Chromatic Index ◽

Proper Edge Coloring ◽

Minimum Number

The adjacent vertex distinguishing edge coloring of a graph [Formula: see text] is a proper edge coloring of [Formula: see text] such that the color sets of any pair of adjacent vertices are distinct. The minimum number of colors required for an adjacent vertex distinguishing edge coloring of [Formula: see text] is denoted by [Formula: see text]. It is observed that [Formula: see text] when [Formula: see text] contains two adjacent vertices of degree [Formula: see text]. In this paper, we prove that if [Formula: see text] is a planar graph without 3-cycles, then [Formula: see text]. Furthermore, we characterize the adjacent vertex distinguishing chromatic index for planar graphs of [Formula: see text] and without 3-cycles. This improves a result from [D. Huang, Z. Miao and W. Wang, Adjacent vertex distinguishing indices of planar graphs without 3-cycles, Discrete Math. 338 (2015) 139–148] that established [Formula: see text] for planar graphs without 3-cycles.

Download Full-text